Procedures Built from Conceptual Understanding

A few weeks ago, I saw a post of some students dancing and singing to a set of procedures for solving a long division problem. The person who shared the video raved about how she had never seen students love math so much. Several of my friends responded by saying that they didn’t love math, they loved singing and dancing! There was another aspect of this video that was not mentioned that may have an even more damaging effect on the learning of mathematics by these students; they were learning a rote procedure. Nothing in the song or the explanations gave any indication that they understood why the steps they used worked.

The National Council of Teachers of Mathematics (NCTM) has published a small book, Principles to Action, that discusses eight teaching practices which promote mathematical development in students. These practices support the new state math standards in a narrow context ,and deep understanding and appreciation of mathematics in a broader context. One of the teaching practices is to build procedural fluency from conceptual understanding. When teachers teach a memorized set of steps without conceptual understanding the learning is disconnected to other math they have learned and does not lay a foundation for math they could be learning in the future. It teaches students that math does not make sense. We refer to procedures of this nature as empty procedures.

In our work with students at the AIMS Center, we came across a student who had learned one empty procedure. Over my next few blog posts, I would like to talk about her as a way of continuing to understand how students come to understand number. This story will serve as an example of the problems that can be created when we build procedures without the conceptual understanding that supports them. This is the story of Chloe.

We begin our story of Chloe as a 1st grader with an experience in May 2015. Chloe would use her fingers to solve tasks like combining 6 and 3, when not given counters to solve the problem. She would show six fingers and then make a three with the other hand and count all the fingers, arriving at the answer 8.

In December 2015 she had adapted her adding strategy to solve 6 + 3 by simultaneously lifting six fingers and then sequentially lifting three more fingers. She answered 9, but when asked, “Nine what?” She said 93.

When solving problems where she didn’t have enough fingers like 8 + 4, she attempted to use her finger pattern strategy, lifting simultaneous fingers for eight and then sequentially lifting four fingers and counting “1, 2, 3, 4”. She was left with seven fingers lifted and answered seven.

In January 2016, when given the problem 8 + 5, Chloe lifted five fingers simultaneously for the second addend and counted-on “9, 10, 11, 12, 13.” When asked about how she figured this out she described putting the first number in her head and counting the second number out.

Someone might think that this was a big improvement and for this problem she did get the correct answer, but in other tasks we gave her she was not able to use this strategy. For her the strategy was just a set of steps to get an answer. We knew this because of how she had recently solved problems and by the limitation of her strategy in other tasks.

Next time, we will further explore the implications for Chloe in other tasks and the connection to the other teaching practices from NCTM.

(Special thanks to Beverly Ford for her assistance in writing this blog.)